Problem 12 - Algebra and Trigonometry

Question

In Exercises 1–14, express each interval in set-builder notation and graph the interval on a number line. $$ (-\infty, 2) $$

Step-by-Step Solution

Answer

In set-builder notation, the given interval \(-\infty, 2\) is represented as \(x : x < 2\). The number line graphing of this interval involves shading the line from far left to just before the point 2 and marking a small open circle at point 2 to indicate that 2 is not part of the set.

Step 1: Convert to Set-builder Notation

In set-builder notation, this interval is represented as \(x : x < 2\). This means that the set of all numbers \(x\) such that \(x\) is less than 2.

Step 2: Draw a Number Line

Draw a horizontal line and denote different points. This line represents the number line. Ensure that there are some points to the left of 2 and some points to the right of 2.

Step 3: Graph the Interval on the Number Line

As the interval \(-\infty, 2\) denotes numbers from negative infinity up to, but not including, 2, you should shade the number line starting from a point far to the left (to represent \(-\infty\)) and stop just before reaching point 2. Then, make an open circle around point 2 to represent that 2 is not included in the interval.

Key Concepts

Interval Notation

Interval notation is a mathematical shorthand that simplifies the way we express a range of numbers. It consists of two numbers separated by a comma and enclosed within brackets or parentheses. The type of bracket used indicates whether the endpoints are included in the interval. For instance, \(a, b\) denotes all numbers greater than \(a\) and less than \(b\), excluding both \(a\) and \(b\), whereas \[a, b\] includes both endpoints.
In the provided example, the interval \( -\infty, 2\) represents all numbers less than 2. Since infinity is a concept, not a number, we always use a parenthesis with \/\infty\ to signify that it is not a specific point on the number line. The interval’s notation indicates that it extends indefinitely in the negative direction but stops just before reaching the number 2, which is not part of the set. Therefore, for the interval \( -\infty, 2\), we'd write in set-builder notation as \(x | x < 2\). This concisely captures all numbers fulfilling the condition \(x < 2\) without listing them out individually.
Using interval notation is very efficient in mathematics, especially when dealing with continuous sets of numbers, as it easily communicates the main characteristics of an interval with minimal symbols.

Number Line Graphing

Graphing an interval on a number line visualizes the set of numbers it includes. It's a crucial skill in understanding ranges and can bring clarity to complex concepts. To graph the interval \( -\infty, 2\) on a number line:
- Begin by drawing a long, horizontal line on your paper or screen. This represents the number line which is infinite in both directions.- Mark a point on the line where the number 2 would be located. Label this point for clarity.- Since 2 is not included in the interval (denoted by the parenthesis), draw an open circle around this point. An open circle indicates exclusion.- To represent the continuation to negative infinity, shade or draw a line to the left of 2, extending towards the end of the number line in the negative direction.- The shaded portion represents all the numbers that are less than 2 and is a visual representation of the set \(x | x < 2\).
By practicing number line graphing, students can better understand intervals and the relationships between numbers, fostering a visual learning approach that can be incredibly beneficial in mastering inequalities and other mathematical concepts.

Inequalities

Inequalities are statements about the relative size or order of two values. They use symbols such as \( < \), \( > \), \(( \leq \)), and \(( \geq \)) to convey that one number is less than, greater than, less than or equal to, or greater than or equal to another number, respectively. In the example of the interval \( -\infty, 2\), the inequality \(x < 2\) represents all numbers that are less than 2. This is an essential concept in algebra and appears frequently across mathematical disciplines.
Understanding inequalities allows us to solve problems that involve ranges of values rather than exact numbers. For instance:

• Identifying solution sets for algebraic equations.
• Describing domain and range for functions.
• Optimizing outcomes in calculus or economics.

Inequalities can be graphed on number lines, as shown in the exercise, or used in conjunction with set-builder notation to neatly express complex sets of numbers. When one grasps the concepts of inequalities, they unlock a deeper understanding of many mathematical principles and problem-solving strategies.